Addendum: Some Limit Theorems of Renewal Theory

1984 ◽  
Vol 28 (4) ◽  
pp. 856-857
Author(s):  
A. V. Nagaev
1984 ◽  
Vol 16 (04) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.


1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.


1984 ◽  
Vol 16 (4) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.


2016 ◽  
Vol 21 (0) ◽  
Author(s):  
Kenneth S. Alexander ◽  
Quentin Berger

1994 ◽  
Vol 116 (1) ◽  
pp. 179-190
Author(s):  
Laurence A. Baxter ◽  
Linxiong Li

AbstractA random environment is modelled by an arbitrary stochastic process, the future of which is described by a σ-algebra. Renewal processes and alternating renewal processes are defined in this environment by considering the conditional distributions of random variables generated by the processes with respect to the σ-algebra. Generalizations of several of the standard limit theorems of renewal theory are derived.


1974 ◽  
Vol 6 (1) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.


2016 ◽  
Vol 38 (1) ◽  
pp. 155-179 ◽  
Author(s):  
PETER HAÏSSINSKY ◽  
PIERRE MATHIEU ◽  
SEBASTIAN MÜLLER

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as analigned unionof independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.


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